Determination of electron-hole correlation length in CdSe quantum dots using explicitly correlated two-particle cumulant
DDetermination of electron-hole correlation length in CdSe quantum dots using explicitly correlatedtwo-particle cumulant
Christopher J. Blanton and Arindam Chakraborty ∗ Department of Chemistry, Syracuse University, Syracuse, New York 13244 (Dated: August 29, 2018)
ABSTRACT:
The electron-hole correlation length serves as an intrin-sic length scale for analyzing excitonic interactions in semiconductornanoparticles. In this work, the derivation of electron-hole correlationlength using the two-particle reduced density is presented. The correla-tion length was obtained by first calculating the electron-hole cumulantfrom the pair density, and then transforming the cumulant into intracu-lar coordinates, and finally then imposing exact sum-rule conditions onthe radial integral of the cumulant. The excitonic wave function for thecalculation was obtained variationally using the electron-hole explicitlycorrelated Hartree-Fock method. As a consequence, both the pair densityand the cumulant were explicit functions of the electron-hole separationdistance. The use of explicitly correlated wave function and the integralsum-rule condition are the two key features of this derivation. The methodwas applied to a series of CdSe quantum dots with diameters 1-20 nm andthe effect of dot size on the correlation length was analyzed.
Keywords: explicitly correlated, Gaussian-type geminal, electron-hole correlation, reduced density matrix, cumulant, transi-tion density matrix
I. Introduction
Electron-hole excitations in semiconductor quantum dotsare influenced by their size, shape and chemical composition.Controlling the generation and the dissociation of electron-hole (eh) pairs have important technological applications inthe field of light-harvesting materials[1–4], photovoltaics[5–8], solid-state lighting[9–12] and lasing[13–16]. In order tocontrol the generation and dissociation of the eh-pair, it is im-portant to understand the underlying interaction between thequasiparticles. Theoretical treatment of electron-hole inter-action in quantum dots is challenging because of the com-putational bottleneck associated with quantum mechanicaltreatment of many-electron systems. In principle, a sim-plified description of the electron-hole pair can be achievedby ignoring the eh interaction and treating them as inde-pendent quasiparticles. Although this approach can dra-matically reduce the computational cost, such simplificationcan lead to qualitatively wrong results. For example, op-tical spectra calculation using independent quasiparticle ap-proach often shows significant deviation from the experimen-tal results. One of the main limitations of the independentquasiparticle method is its inability in describing bound ex-citonic states. Multiexcitonic interaction, exciton and biex-citon binding energies, radiative and Auger recombination ∗ corresponding author: [email protected] are some of the properties whose calculations depend onthe accurate treatment of electron-hole correlation. Theoret-ical investigation of electron-hole correlation has been per-formed using various methods such as time-dependent densityfunctional theory (TDDFT)[17–24], perturbation theory[25],GW combined with Bethe-Salpeter equation[26–34], con-figuration interaction[35–43], quantum Monte Carlo[38, 44–46], path-integral Monte Carlo,[47, 48] explicitly correlatedHartree-Fock method,[49–53] and electron-hole density func-tional theory.[54]In this work, we are interested in the calculation of electron-hole correlation length (eh-CL) in CdSe quantum dots. Ourgoal is to provide a statistical definition of the electron-hole correlation length. The concept of correlation lengthhas been widely used in many fields, including statisticalmechanics[55–58] and polymer science.[56–62] One of theimportant features of the eh-CL is that is it provides an in-trinsic length scale for describing the electron-hole interac-tion. Because of this, it can play an important role in describ-ing excitonic effects in quantum dots and other nanomaterialssuch as carbon nanotubes.[63–65] The eh-CL can also be usedfor construction of electron-hole correlation functional formulticomponent density functional theory.[54] For example,Salahub and co-workers have developed a series of exchange-correlation functions that are based on electron-electron cor-relation length[66–69] and a similar strategy can be used forconstruction of electron-hole correlation functionals using eh-CL. The eh-CL can also aid in the development of explicitlycorrelated wave functions (such as Jastrow and Gaussian-typegeminal functions) which depend directly on the electron-holeseparation distance.[46, 49–53]We have used the 2-particle electron-hole density matrix a r X i v : . [ phy s i c s . a t m - c l u s ] M a y Electron-hole cumulant II THEORY for the definition and calculation of the eh-CL. Two-particlereduced density matrix (2-RDM) has been used extensivelyfor investigation of electron-electron correlation[71–77] andelectronic excitation[78] in many-electron systems. For thepresent system, the 2-RDM is the appropriate mathemati-cal quantity that contains all the necessary information aboutelectron-hole correlation. Specifically, the cumulant associ-ated with the electron-hole 2-RDM is the component of the2-RDM that cannot be expressed as a product of 1-particleelectron and hole densities. In principle, the 2-RDM can beobtained directly without the need for an underlying wavefunction as long as the N -representability of 2-RDM can besatisfied. However, in the present work, we have obtainedthe 2-RDM from an explicitly correlated electron-hole wavefunction. The remainder of the article is organized as follows.The derivation of eh-CL from the electron-hole cumulant ispresented in subsection II A, transformation to intracular andextracular coordinates is described in subsection II B, and de-tails of the explicitly correlated electron-hole wave functionare presented in subsection II C and section III. The methodwas applied to a series of CdSe quantum dots and the resultsare presented in section IV. II. Theory
A. Electron-hole cumulant
The interaction between the quasiparticles in the quantumdot is described the electron-hole Hamiltonian[36–38, 46, 49,50, 52, 53, 80–87] which has the following general expression H = ∑ i j (cid:104) i | − ¯ h m e ∇ + v eext | j (cid:105) e † i e j (1) + ∑ i j (cid:104) i | − ¯ h m h ∇ + v hext | j (cid:105) h † i h j + ∑ i ji (cid:48) j (cid:48) (cid:104) i ji (cid:48) j (cid:48) | ε − r − | i ji (cid:48) j (cid:48) (cid:105) e † i e j h † i (cid:48) h j (cid:48) + ∑ i jkl w ee i jkl e † i e † j e l e k + ∑ i jkl w hh i jkl h † i h † j h l h k . We define the electron-hole wave function for a multiexci-tonic system consisting of N e and N h number of electrons andholes, respectively by Ψ eh ( x e1 , . . . , x e N e , x h1 , . . . , x h N h ) , where x isa compact notation for both the spatial and spin coordinate ofthe particles. The spin-integrated 2-particle reduced densitycan be obtained from the electron-hole wave function by inte-gration over the N e − N h − ρ eh ( r e , r h ) = N e N h (cid:104) Ψ eh | Ψ eh (cid:105) (cid:90) ds e1 ds h1 d x e2 , . . . , x e N e x h2 , . . . , x h N h Ψ ∗ eh Ψ eh (2)where, integration over the spin coordinate s is performed forboth electron and hole. The single-particle density is obtainedfrom the 2-particle density using the sum-rule condition[70] ρ e ( r e ) = N h (cid:90) d r h ρ eh ( r e , r h ) , (3) ρ h ( r h ) = N e (cid:90) d r e ρ eh ( r e , r h ) . (4)We define the electron-hole cumulant as the difference be-tween the 2-particle density and the product of the 1-particleelectron and hole densities as shown in the following equation q ( r e , r h ) = ρ eh ( r e , r h ) − ρ e ( r e ) ρ h ( r h ) . (5)This definition is analogous to the definition used by Mazz-iotti et al.[88] in electronic structure theory. By construction,the cumulant contains information about correlation betweenthe two particles. Consequently, the Coulomb contribution ofthe electron-hole correlation energy can be directly expressedin terms of the electron-hole cumulant and is given by the fol-lowing expression (cid:104) Ψ eh | V eh | Ψ eh (cid:105) = (cid:104) ρ eh ε − r − (cid:105) (6) = J eh + (cid:104) q ( r e , r h ) ε − r − (cid:105) , where ε is the dielectric constant and J eh is the classicalCoulomb electron-hole energy J eh = (cid:104) ρ e ρ h ε − r − (cid:105) . (7)The cumulant has an important property that its integrationover all space should be zero due to the density sum-ruleconditions[70] (cid:90) d r e d r h q ( r e , r h ) = . (8)We use this relationship for the definition of the electron-holecorrelation length. B. Intracular and extracular coordinates
Beginning with Coleman’s initial definition of the intrac-ule and extacule matrices in terms of the center of mass(extracule) and relative motion (intracule) coordinates,[89]the concept of the intracule and extracule in the regime ofelectronic systems has been previously explored in earlierstudies.[89–95] The intracular and extracular coordinates forthe eh-system are defined by r eh = r e − r h (9) R = (cid:16) r e + r h (cid:17) . (10)The integral of the cumulant is expressed in terms of thesecoordinates (cid:90) d r e d r h q ( r e , r h ) = (cid:90) d r eh (cid:90) d R q ( r eh , R ) (11) = (cid:90) ∞ dr eh r (cid:90) d Ω sin θ (cid:90) d R q ( r eh , R ) (12) = (cid:90) ∞ dr eh r q r ( r eh ) (13)2 Explicitly correlated electron-hole wave function II THEORY
In the above expression, the integral over the intracular coor-dinate r eh is transformed into spherical polar coordinates. Thefunction q r is the spherically averaged radial cumulant and theintegral of the radial cumulant over a finite limit is used to de-fine the following function I ( d ) I ( d ) = (cid:90) d dr eh r q r ( r eh ) . (14)The zero-integral property of q (defined in Eq. (8)) ensuresthat this integral goes to zero at large d lim d → ∞ I ( d ) = . (15)Here, we use I ( d ) to define the electron-hole correlationlength. Specifically, the electron-hole correlation length ( r c )is defined as the value of d at which the value of I ( d ) is zero | I ( r c ) | = r c << ∞ . (16)The description of the electron-hole wave function used forthe calculation of the radial cumulant is presented in the fol-lowing section. C. Explicitly correlated electron-hole wave function
We have used the electron-hole explicitly correlatedHartree-Fock method (eh-XCHF) for obtaining the electron-hole wave function. This method has been used in earlier workfor the computation of exciton binding energies and electron-hole recombination probabilities in quantum dots.[49–53] Abrief summary of the eh-XCHF method is presented here andthe implementation details of this method can be found inwork by Elward and co-workers.[50–52] The ansatz of the eh-XCHF wave function consists of multiplying the mean-fieldelectron-hole reference wave functions with an explicitly cor-related function G as shown in the following equation Ψ eh − XCHF = G Φ e Φ h , (17)where G is the geminal operator G = N e ∑ i = N h ∑ j = g ( r i j ) , (18) g ( r eh ) = N g ∑ k = b k exp ( − γ k r ) . (19)The eh-XCHF method is a variational method in which thecorrelation function G and the reference wave function are ob-tained by minimizing the total energy E eh − XCHF = min G , Φ e , Φ h (cid:104) G Φ | H | G Φ (cid:105)(cid:104) G Φ | G Φ (cid:105) , (20)where Φ = Φ e Φ h . To perform the above minimization,it is more efficient to work with the following congruent-transformed operators ˜ H = G † HG , (21) ˜1 = G † G . (22)This transformation is particularly important for the calcula-tion of the 2-particle reduced density matrix in the presentwork. The set of parameters { b k , γ k } in G were obtained bynon-linear optimization, and for a given set of these parame-ters, the minimization over the reference wave function wasperformed by determining the self-consistent solution of thecoupled Fock equations˜ F e C e = ˜ S e C e λ e , (23)˜ F h C h = ˜ S h C h λ h . (24)The tilde in the above expressions represent that the Fock andthe overlap matrices incorporate the transformed operators de-fined in Eq. (21).The transformed operator ˜1 can be written as a sum of op-erators as shown below˜1 = G † G (25) = ∑ ii (cid:48) g ( i , i (cid:48) ) ∑ j j (cid:48) g ( j , j (cid:48) ) ( i , j = , . . . , N e ; i (cid:48) , j (cid:48) = , . . . , N h ) (26) = ∑ ii (cid:48) g ( i , i (cid:48) ) g ( i , i (cid:48) ) + ∑ i (cid:54) = j , i (cid:48) g ( i , i (cid:48) ) g ( j , i (cid:48) ) (27) + ∑ i (cid:48) (cid:54) = j (cid:48) , i g ( i , i (cid:48) ) g ( i , j (cid:48) ) + ∑ i (cid:54) = j , i (cid:48) (cid:54) = j (cid:48) g ( i , i (cid:48) ) g ( j , j (cid:48) ) . The above expression can be written in a compact notation asa sum of 2, 3, and 4-particle operators G † G = Ω + Ω + Ω + Ω . (28)The 2-particle density for the eh-XCHF wave function can beexpressed in terms of these operators as shown below ρ eh ( r e , r h ) = N e N h (cid:104) Ψ eh − XCHF | Ψ eh − XCHF (cid:105) (29) × (cid:104) Ψ ∗ eh − XCHF Ψ eh − XCHF (cid:105) s , s (cid:48) , , (cid:48) ,..., N e , N h , where the subscript in the above expression is a compact nota-tion for integration over the remaining coordinates describedin Eq. (2). Substituting the expression from Eq. (28), we getthe following expression ρ eh ( r e , r h ) = N e N h (cid:104) Φ | ˜1 | Φ (cid:105) (30) × (cid:104) Φ ∗ ( Ω + Ω + Ω + Ω ) Φ (cid:105) s , s (cid:48) , , (cid:48) ,..., N e , N h . For a multiexcitonic system all 2, 3, and 4-particle operatorsshould be used for the computation of the 2-particle density.In a related work on many-electron system, we have shownthat it is possible to avoid integration over higher-order op-erators by using diagrammatic summation technique and asimilar strategy can be used for multiexcitonic systems aswell.[96]3
II COMPUTATIONAL DETAILS
D. Relation to uncorrelated transition density matrices
One of the important features of the correlation function isthat it allows for a compact representation of the 2-particledensity matrix in the position representation. The relationshipcan be readily seen by expanding the eh-XCHF wave functionin the Slater determinant basis G Φ = ∞ ∑ ii (cid:48) (cid:104) Φ e i Φ h i (cid:48) | G | Φ e0 Φ h0 (cid:105) (cid:124) (cid:123)(cid:122) (cid:125) c ii (cid:48) Φ e i Φ h i (cid:48) = ∞ ∑ ii (cid:48) c ii (cid:48) Φ e i Φ h i (cid:48) . (31)Substituting Eq. (31) in the expression of ρ eh gives ρ eh ( r e , r h ) = N e N h (cid:104) Φ | ˜1 | Φ (cid:105) (32) × (cid:104) ∞ ∑ i j ∞ ∑ i (cid:48) j (cid:48) c ∗ ii (cid:48) c j j (cid:48) Φ e ∗ i Φ h ∗ i (cid:48) Φ e j Φ h j (cid:48) (cid:105) s , s (cid:48) , , (cid:48) ,..., N e , N h = N e N h (cid:104) Φ | ˜1 | Φ (cid:105) ∞ ∑ i j ∞ ∑ i (cid:48) j (cid:48) c ∗ ii (cid:48) c j j (cid:48) d e i j d h i (cid:48) j (cid:48) , (33)where the transition density matrix d i j is defined as d e i j ( r e ) = (cid:104) Φ e ∗ i Φ e j (cid:105) s , ,..., N e . (34)It is seen from Eq. (32) that the 2-particle density obtainedfrom the eh-XCHF wave function is equivalent to the infinite-order expansion in terms of the transition density matrices. III. Computational details
The method described in section II was used for cal-culating electron-hole correlation length in CdSe quantumdots in the range of 1-20 nm in diameter. We are inter-ested in the effect of dot size on the electron-hole correla-tion length for a single electron-hole pair in CdSe quantumdots. For a single electron-hole pair, the higher-order op-erators in Eq. (30) rigorously vanish from the expression.This provides considerable simplification in the calculationof the 2-particle density. Because of the dot size, appli-cation of either DFT or atom-centered pseudopotential ap-proach is computationally prohibitive. To make the compu-tation tractable, we have used a parabolic confining poten-tial in the electron-hole Hamiltonian described in Eq. (1).Parabolic confinement potential in quantum dots has beenused extensively for various properties such as total exci-ton energy[97, 98], exciton dissociation[99], exciton bind-ing energy[49, 50, 52, 100] eh-recombination probability[49–51], effect of magnetic[101–106] and electric fields[53, 101,107–109], exciton-polariton condensate[110], linear opticalproperties[111, 112], optical rectification[113], non-linearrectification[107], dynamics[114], eh-correlation energy[115,116], resonant tunneling[117], collective modes[118], andthermodynamic properties[119]. The external potential for theelectron and hole quasiparticle was defined as v ext α = k α | r α | α = e , h (35) TABLE I.
Force constants for CdSe quantum dots.
Dot diameter (nm) k e (atomic units) k h (atomic units)1 .
24 2 . × − . × − .
79 6 . × − . × − .
76 1 . × − . × − .
98 8 . × − . × − .
28 5 . × − . × − .
79 3 . × − . × − .
80 1 . × − . × − .
00 1 . × − . × − .
60 3 . × − . × − .
00 6 . × − . × − .
00 1 . × − . × − .
00 4 . × − . × − where k α is the force constant which determines the strengthof the confinement potential. We have used a particle-numberbased search procedure for determination of the force constant k α . The central idea of this approach is to find the value of k α such that the computed 1-particle electron and hole densitiesare confined within the volume of the quantum dot. This isobtained by performing the following minimizationmin k min α (cid:32) N α − (cid:90) D dot2 drr (cid:90) d Ω ρ α ( r ) (cid:33) , (36)where D dot is the diameter of the quantum dot and Ω is theangular coordinate. The values of the force constants usedfor each dot is listed in Table I. The kinetic energy op-erator was computed using the electron and hole effectivemasses of 0 .
13 and 0 .
38 atomic units, respectively.[81] Theinteraction between the electron and hole was described byscreened Coulomb potential. We have used the size and dis-tance dependent dielectric function ε ( r , R dot ) , which was de-veloped by Wang and Zunger for CdSe.[120] The electronand hole molecular orbitals in Φ were represented using alinear combination of Gaussian type orbitals (GTOs) and theexpansion coefficients were obtained by the solving the cou-pled Fock equations shown in Eq. (23). The basis used wasa single S Cartesian GTO was used and the exponents ofthe basis functions are listed in Table II. The use of GTOsis especially convenient because the integrals involving theGTOs and the Gaussian correlation function, G , are knownanalytically.[121–124] For a given value of r e , the 1-particledensity ρ was calculated analytically. The integration over theintracular coordinate in Eq. (14) was performed numerically.The correlation function, G , was expanded as a linear com-bination of six Guassian-type geminal functions[49, 50, 52]and the set of { b k , γ k } parameters were optimized for each dotsize. The first set of geminal parameters was set to b = γ = CONCLUSIONS
TABLE II.
Exponent used in GTO basis e − α r . Dot diameter (nm) α (atomic units)1 .
24 2 . × − .
78 1 . × − .
76 5 . × − .
98 5 . × − .
28 4 . × − .
79 3 . × − .
80 1 . × − .
00 1 . × − .
60 1 . × − .
00 4 . × − .
00 2 . × − .
00 1 . × − IV. Results
The electron-hole correlation length was obtained by inte-gration of the radial cumulant as described in Eq. (14). InFigure 1, the integral of the cumulant, I ( d ) , for three differentdot sizes are presented. As expected, the integral goes to zero FIG. 1. The value of I ( d ) as d , the upper limit in Eq. (14), is variedfor the 1.78 nm, 6.6 nm, and 20 nm diameter CdSe quantum dots. at large distances (high d values) and the distance at whichthe integral converges to zero is defined as the electron-holecorrelation length r c . The calculated electron-hole correla-tion lengths are presented in Table IV. We find that, in allcases, the correlation length increases with increasing dot di-ameter. Another quantity that is important for investigatingelectron-hole correlation is the length scale associated withthe first node of the radial cumulant. We define this quan-tity as r node and the calculated values are presented in Ta-ble IV. The maximum of the I ( d ) in Figure 1 corresponds to r node . Because the interaction between the electron and holeis attractive, we expect an enhancement in the pair density ascompared to mean-field density at small r eh distances. This phenomenon is opposite to the correlation hole observed inelectron-electron interaction, in which small r ee shows a de-crease in correlated electron-pair density as compared to un-correlated electron density. The r node can be interpreted asthe effective radius of the sphere that encloses the region ofenhanced probability density. As seen from Table IV, r c and r node are similar in magnitude for small dot sizes, but thesequantities differ significantly for larger dots. The correlationlength as a function of the dot diameter is plotted in Figure 2.The set of data showed good agreement with the linear fit,with a mean absolute error of 0.323 nm. A trend of increasingcorrelation length with increasing dot diameter is observed.The correlation lengths show that correlation effects are im-portant even at long electron-hole separations. The linear re- FIG. 2. The correlation length as a function of the dot diameter D dot .The linear fit is r C = . D dot , with a mean absolute error of 0.323nm. lationship between the dot diameter and the correlation lengthhas an important application in the construction of compactexplicitly correlated electron-hole wave function. For exam-ple, the determination of the eh-XCHF wave function requiresthe optimization of the non-linear parameters { b k , γ k } in G .By using the relationship between the dot diameter and cor-relation length, it is possible to assign the non-linear parame-ters as some multiple of the correlation length. This approachavoids optimization of non-linear parameters and can result insignificant reduction in the computational effort. V. Conclusions
In conclusion, we have presented a method for calculatingelectron-hole correlation length in semiconductor quantumdots. We have used the cumulant derived from the electron-hole 2-particle density as the central quantity for defining thecorrelation length. There are two key features of this method.First, the 2-particle reduced density was obtained from an ex-plicitly correlated electron-hole wave function. Consequently,the reduced density matrix and the corresponding cumulant5
CONCLUSIONS
TABLE III.
Value of the geminal parameters for CdSe quantum dots.
Dot Diameter (nm) b γ b γ b γ b γ b γ .
25 1 . × − . × − . × − . × − . × − . × . × − . × − . × − . × − .
78 1 . × − . × − . × − . × − . × − . × − . × − . × . × − . × − .
76 2 . × − . × − . × − . × − . × − . × − − . × − . × − . × − . × .
98 2 . × − . × − . × − . × − . × − . × − . × − . × . × − . × − .
28 2 . × − . × − . × − . × − . × − . × − . × − . × − . × − . × .
79 2 . × − . × − . × − . × − . × − . × − . × − . × − . × − . × .
80 3 . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − .
00 3 . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − .
60 4 . × − . × − . × − . × − . × − . × . × − . × − . × − . × − .
00 5 . × − . × − . × − . × − . × − . × . × − . × − . × − . × − .
00 6 . × − . × − . × . × − . × − . × . × − . × − . × − . × − .
00 7 . × − . × − . × . × − . × − . × − . × − . × − . × − . × − TABLE IV.
Electron-hole correlation lengths and r node for CdSequantum dots. Dot Diameter Correlation length r node (nm) (nm) (nm)1 .
24 0 .
381 0 . .
78 0 .
683 0 . .
76 1 .
905 0 . .
98 2 .
117 0 . .
28 2 .
572 0 . .
79 2 .
778 0 . .
80 3 .
293 1 . .
00 3 .
307 1 . .
60 4 .
047 1 . .
00 6 .
156 2 . .
00 10 .
164 3 . .
00 11 .
930 4 . were explicit functions of the electron-hole separation dis-tance. Second, the calculation of the correlation length wasnot based on the nodes of the cumulant but was derived fromthe exact sum rule relationship satisfied by all N -representablecumulants. The developed method was applied to a series ofCdSe quantum dots and a linear relationship between the dotsize and correlation length was observed. The electron-holecorrelation length provides a natural length scale for investi-gating electron-hole correlation in nanoparticles. We envisionthat in future work, the electron-hole correlation length willbe used in the construction of compact explicitly correlatedwave functions and also for developing multi-component[54]electron-hole density functionals. Acknowledgments
We wish to thank ACS-PRF grant 52659-DNI6 and Syra-cuse University for financial support. [1] Haiming Zhu, Nianhui Song, and Tianquan Lian, “Wavefunction engineering for ultrafast charge separation and slowcharge recombination in type ii core/shell quantum dots,”Journal of the American Chemical Society , 8762–8771(2011).[2] Kaifeng Wu, Zheng Liu, Haiming Zhu, and Tianquan Lian,“Exciton annihilation and dissociation dynamics in group iivcd3p2 quantum dots,” The Journal of Physical Chemistry A , 6362–6372 (2013).[3] Michael Gr¨atzel, “Solar energy conversion by dye-sensitizedphotovoltaic cells,” Inorganic Chemistry , 6841–6851(2005), pMID: 16180840.[4] Prashant V. Kamat, “Quantum dot solar cells. semiconduc-tor nanocrystals as light harvesters,” The Journal of PhysicalChemistry C , 18737–18753 (2008).[5] Zhen Li, Libo Yu, Yingbo Liu, and Shuqing Sun, “Cds/cdsequantum dots co-sensitized tio2 nanowire/nanotube solar cellswith enhanced efficiency,” Electrochimica Acta , 379 –388 (2014).[6] Mohammad Reza Golobostanfard and Hossein Abdizadeh,“Tandem structured quantum dot/rod sensitized solar cellbased on solvothermal synthesized cdse quantum dots and rods,” Journal of Power Sources , 102 – 109 (2014).[7] S.A. Mcdonald, G. Konstantatos, S. Zhang, P.W. Cyr, E.J.D.Klem, L. Levina, and E.H. Sargent, “Solution-processed pbsquantum dot infrared photodetectors and photovoltaics,” Na-ture Materials , 138–142 (2005).[8] Tridip Ranjan Chetia, Dipankar Barpuzary, and MohammadQureshi, “Enhanced photovoltaic performance utilizing effec-tive charge transfers and light scattering effects by the combi-nation of mesoporous, hollow 3d-zno along with 1d-zno in cdsquantum dot sensitized solar cells,” Phys. Chem. Chem. Phys. , 9625–9633 (2014).[9] Polina O. Anikeeva, Jonathan E. Halpert, Moungi G. Bawendi,and Vladimir Bulovc, “Electroluminescence from a mixedred-green-blue colloidal quantum dot monolayer,” Nano Let-ters , 2196–2200 (2007), pMID: 17616230.[10] Eunjoo Jang, Shinae Jun, Hyosook Jang, Jungeun Lim,Byungki Kim, and Younghwan Kim, “White-light-emittingdiodes with quantum dot color converters for display back-lights,” Advanced Materials , 3076–3080 (2010).[11] In Seong Sohn, Sanjith Unithrattil, and Won Bin Im, “Stackedquantum dot embedded silica film on a phosphor plate for su-perior performance of white light-emitting diodes,” ACS Ap- CONCLUSIONS plied Materials & Interfaces , 5744–5748 (2014).[12] Lauren E. Shea-Rohwer, James E. Martin, Xichen Cai, andDavid F. Kelley, “Red-emitting quantum dots for solid-statelighting,” ECS Journal of Solid State Science and Technology , R3112–R3118 (2013).[13] V. I. Klimov, A. A. Mikhailovsky, Su Xu, A. Malko, J. A.Hollingsworth, C. A. Leatherdale, H.-J. Eisler, and M. G.Bawendi, “Optical gain and stimulated emission in nanocrys-tal quantum dots,” Science , 314–317 (2000).[14] C. Foucher, B. Guilhabert, N. Laurand, and M. D. Dawson,“Wavelength-tunable colloidal quantum dot laser on ultra-thinflexible glass,” Applied Physics Letters , 141108 (2014).[15] Yuchang Wu and Levon V. Asryan, “Direct and indirect cap-ture of carriers into the lasing ground state and the light-current characteristic of quantum dot lasers,” Journal of Ap-plied Physics , 103105 (2014).[16] Yue Wang, Van Duong Ta, Yuan Gao, Ting Chao He, RuiChen, Evren Mutlugun, Hilmi Volkan Demir, and Han DongSun, “Stimulated emission and lasing from cdse/cds/zns core-multi-shell quantum dots by simultaneous three-photon ab-sorption,” Advanced Materials , 2954–2961 (2014).[17] Giuliano Malloci, Letizia Chiodo, Angel Rubio, and Alessan-dro Mattoni, “Structural and optoelectronic properties of un-saturated zno and zns nanoclusters,” The Journal of PhysicalChemistry C , 8741–8746 (2012).[18] A. Castro, J. Werschnik, and E. K. U. Gross, “Controlling thedynamics of many-electron systems from first principles: Acombination of optimal control and time-dependent density-functional theory,” Phys. Rev. Lett. , 153603 (2012).[19] C. A. Ullrich and G. Vignale, “Collective charge-density exci-tations of noncircular quantum dots in a magnetic field,” Phys.Rev. B , 2729–2736 (2000).[20] Sean A. Fischer, Angela M. Crotty, Svetlana V. Kilina,Sergei A. Ivanov, and Sergei Tretiak, “Passivating ligand andsolvent contributions to the electronic properties of semicon-ductor nanocrystals,” Nanoscale , 904–914 (2012).[21] Kim Hyeon-Deuk and Oleg V. Prezhdo, “Multiple excitongeneration and recombination dynamics in small si and cdsequantum dots: An ab initio time-domain study,” ACS Nano ,1239–1250 (2012).[22] Ekaterina Badaeva, Joseph W. May, Jiao Ma, Daniel R.Gamelin, and Xiaosong Li, “Characterization of excited-state magnetic exchange in mn2+-doped zno quantum dots us-ing time-dependent density functional theory,” The Journal ofPhysical Chemistry C , 20986–20991 (2011).[23] Marie Lopez del Puerto, Murilo L. Tiago, and James R. Che-likowsky, “Excitonic effects and optical properties of passi-vated cdse clusters,” Phys. Rev. Lett. , 096401 (2006).[24] M. Claudia Troparevsky, Leeor Kronik, and James R. Che-likowsky, “Optical properties of cdse quantum dots,” The Jour-nal of Chemical Physics , 2284–2287 (2003).[25] Daniel Neuhauser, Eran Rabani, and Roi Baer, “Expeditiousstochastic approach for mp2 energies in large electronic sys-tems,” Journal of Chemical Theory and Computation , 24–27(2013).[26] V Janis and V Pokorny, “Quantum transport in strongly disor-dered crystals: Electrical conductivity with large negative ver-tex corrections,” Journal of Physics: Conference Series (2012), 10.1088/1742-6596/400/4/042023.[27] G Pal, G Lefkidis, H C Schneider, and W H¨ubner, “Opti-cal response of small closed-shell sodium clusters,” Journal ofChemical Physics (2010), 10.1063/1.3494093.[28] G Pal, Y Pavlyukh, W H¨ubner, and H C Schneider, “Opticalabsorption spectra of finite systems from a conserving Bethe- Salpeter equation approach,” European Physical Journal B ,327–334 (2011).[29] V Perebeinos, J Tersoff, and P Avouris, “Radiative lifetimeof excitons in carbon nanotubes,” Nano Letters , 2495–2499(2005).[30] P Puschnig and C Ambrosch-Draxl, “Optical absorption spec-tra of semiconductors and insulators including electron-holecorrelations: An ab initio study within the LAPW method,”Physical Review B - Condensed Matter and Materials Physics , 1651051–1651059 (2002).[31] Michael Rohlfing and Steven G. Louie, “Electron-hole excita-tions in semiconductors and insulators,” Phys. Rev. Lett. ,2312–2315 (1998).[32] Yun-Feng Jiang, Neng-Ping Wang, and Michael Rohlfing,“Quasiparticle band structure and optical spectrum of libr,”The European Physical Journal B , 1–6 (2013).[33] Yuan Ping, Dario Rocca, Deyu Lu, and Giulia Galli, “¡i¿abinitio¡/i¿ calculations of absorption spectra of semiconductingnanowires within many-body perturbation theory,” Phys. Rev.B , 035316 (2012).[34] Michael Rohlfing and Steven G. Louie, “Electron-hole exci-tations and optical spectra from first principles,” Phys. Rev. B , 4927–4944 (2000).[35] M Brasken, S Corni, M. Lindberg, J. Olsen, and D. Sundholm,“Full configuration interaction studies of phonon and photontransition rates in semiconductor quantum dots,” MolecularPhysics: An International Journal at the Interface BetweenChemistry and Physics , 911–918 (2002).[36] S Corni, M Brasken, M Lindberg, J Olsen, and D Sund-holm, “Stabilization energies of charged multiexciton com-plexes calculated at configuration interaction level,” PhysicaE: Low-Dimensional Systems and Nanostructures , 436–442 (2003).[37] S Corni, M Brasken, M Lindberg, J Olsen, and D Sundholm,“Electron-hole recombination density matrices obtained fromlarge configuration-interaction expansions,” Physical ReviewB - Condensed Matter and Materials Physics , 853141–853147 (2003).[38] Y Z Hu, M Lindberg, and S W Koch, “Theory of optically ex-cited intrinsic semiconductor quantum dots,” Physical ReviewB , 1713–1723 (1990).[39] Lixin He, Gabriel Bester, and Alex Zunger, “Singlet-tripletsplitting, correlation, and entanglement of two electrons inquantum dot molecules,” Phys. Rev. B , 195307 (2005).[40] Alberto Franceschetti and Alex Zunger, “Exciton dissociationand interdot transport in cdse quantum-dot molecules,” Phys.Rev. B , 153304 (2001).[41] J. M. An, A. Franceschetti, and A. Zunger, “The excitonic ex-change splitting and radiative lifetime in pbse quantum dots,”Nano Letters , 2129–2135 (2007).[42] Eran Rabani, Balzs Hetnyi, B. J. Berne, and L. E. Brus, “Elec-tronic properties of cdse nanocrystals in the absence and pres-ence of a dielectric medium,” The Journal of Chemical Physics , 5355–5369 (1999).[43] Davood Farmanzadeh and Leila Tabari, “An ab initio study ofthe ground and excited states of mercaptoacetic acid-cappedsilicon quantum dots,” Monatshefte fr Chemie - ChemicalMonthly , 1281–1286 (2013).[44] J Shumway, “Quantum Monte Carlo simulation of exciton-exciton scattering in a GaAs/AlGaAs quantum well,” PhysicaE: Low-Dimensional Systems and Nanostructures , 273–276 (2006).[45] J Shumway and D M Ceperley, “Quantum Monte Carlo sim-ulations of exciton condensates,” Solid State Communications CONCLUSIONS , 19–22 (2005).[46] X Zhu, M S Hybertsen, and P B Littlewood, “Electron-holesystem revisited: A variational quantum Monte Carlo study,”Physical Review B - Condensed Matter and Materials Physics , 13575–13580 (1996).[47] M Harowitz, D Shin, and J Shumway, “Path-integral quantumMonte Carlo techniques for self-assembled quantum dots,”Journal of Low Temperature Physics , 211–226 (2005).[48] M Harowitz and J Shumway, “Path integral simulations ofcharged multiexcitons in InGaAs/GaAs quantum dots,” in PHYSICS OF SEMICONDUCTORS: 27th International Con-ference on the Physics of Semiconductors, ICPS-27 , Vol. 772(2005) pp. 697–698.[49] J M Elward and A Chakraborty, “Effect of dot size on excitonbinding energy and electron-hole recombination probability inCdSe quantum dots,” Journal of Chemical Theory and Com-putation , 4351–4359 (2013).[50] J M Elward, J Hoffman, and A Chakraborty, “Investigation ofelectron-hole correlation using explicitly correlated configura-tion interaction method,” Chemical Physics Letters , 182–186 (2012).[51] J M Elward, J Hoja, and A Chakraborty, “Variational solu-tion of the congruently transformed Hamiltonian for many-electron systems using a full-configuration-interaction calcu-lation,” Physical Review A - Atomic, Molecular, and OpticalPhysics (2012), 10.1103/PhysRevA.86.062504.[52] J M Elward, B Thallinger, and A Chakraborty, “Calculation ofelectron-hole recombination probability using explicitly corre-lated Hartree-Fock method,” Journal of Chemical Physics (2012), 10.1063/1.3693765.[53] C J Blanton, C Brenon, and A Chakraborty, “Developmentof polaron-transformed explicitly correlated full configura-tion interaction method for investigation of quantum-confinedStark effect in GaAs quantum dots,” Journal of ChemicalPhysics (2013), 10.1063/1.4789540.[54] Leonard M. Sander, Herbert B. Shore, and L. J. Sham, “Sur-face structure of electron-hole droplets,” Phys. Rev. Lett. ,533–536 (1973).[55] S G Brush, “History of the Lenz-Ising model,” Reviews ofModern Physics , 883–893 (1967).[56] A. Alan Middleton and Daniel S. Fisher, “Three-dimensionalrandom-field ising magnet: Interfaces, scaling, and the natureof states,” Phys. Rev. B , 134411 (2002).[57] A. Alan Middleton and Ned S. Wingreen, “Collective trans-port in arrays of small metallic dots,” Phys. Rev. Lett. ,3198–3201 (1993).[58] Creighton K. Thomas and A. Alan Middleton, “Exact al-gorithm for sampling the two-dimensional ising spin glass,”Phys. Rev. E , 046708 (2009).[59] C Domb and G S Joyce, “Cluster expansion for a polymerchain,” (1972).[60] P D Gujrati, “A binary mixture of monodisperse polymers offixed architectures, and the critical and the theta states,” Jour-nal of Chemical Physics , 5104–5121 (1998).[61] J L Spouge, “Exact solutions for diffusion-reaction processesin one dimension: II. Spatial distributions,” Journal of PhysicsA: Mathematical and General , 4183–4199 (1988).[62] A Vilenkin, “The theory of melting in heteropolymers. I.Random chains,” Journal of Statistical Physics , 391–404(1978).[63] L Luer, S Hoseinkhani, D Polli, J Crochet, T Hertel, andG Lanzani, “Size and mobility of excitons in (6, 5) carbon-nanotubes,” Nature Physics , 54–58 (2009). [64] S W Koch, W Hoyer, M Kira, and V S Filinov, “Excitonionization in semiconductors,” Physica Status Solidi (B) Ba-sic Research , 404–410 (2003).[65] P Corfdir, B Van Hattem, E Uccelli, S Conesa-Boj, P Lefeb-vre, A Fontcuberta I Morral, and R T Phillips, “Three-dimensional magneto-photoluminescence as a probe of theelectronic properties of crystal-phase quantum disks in GaAsnanowires,” Nano Letters , 5303–5310 (2013).[66] E I Proynov and D R Salahub, “Simple but efficient correlationfunctional from a model pair-correlation function,” PhysicalReview B , 7874–7886 (1994).[67] E I Proynov, A Vela, and D R Salahub, “Gradient-freeexchange-correlation functional beyond the local-spin-densityapproximation,” Physical Review A , Physical Review A ,3766–3774 (1994).[68] E I Proynov, A Vela, and D R Salahub, “Nonlocal correlationfunctional involving the Laplacian of the density,” ChemicalPhysics Letters , 419–428 (1994).[69] E I Proynov, A Vela, and D R Salahub, “Nonlocal correla-tion functional involving the Laplacian of the density (Chem.Phys. Letters 230 (1994) 419) (PII:0009-2614(94)01189-3),”Chemical Physics Letters , 462 (1995).[70] Robert G Parr and Weitao Yang, Density-functional theory ofatoms and molecules , Vol. 16 (Oxford university press, 1989)pp. 32–35.[71] D A Mazziotti, “Quantum chemistry without wave functions:Two-electron reduced density matrices,” Accounts of Chemi-cal Research , 207–215 (2006).[72] D.A. Mazziotti, “Chapter 3: Variational two-electron reduced-density-matrix theory,” Advances in Chemical Physics ,21–59 (2007).[73] D.A. Mazziotti, “Two-electron reduced density matrix asthe basic variable in many-electron quantum chemistry andphysics,” Chemical Reviews , 244–262 (2012).[74] D.R. Rohr, J. Toulouse, and K. Pernal, “Combining density-functional theory and density-matrix-functional theory,” Phys-ical Review A - Atomic, Molecular, and Optical Physics (2010), 10.1103/PhysRevA.82.052502.[75] D.R. Rohr and K. Pernal, “Open-shell reduced density matrixfunctional theory,” Journal of Chemical Physics (2011),10.1063/1.3624609.[76] A.K. Rajam, I. Raczkowska, and N.T. Maitra, “Semiclas-sical electron correlation in density-matrix time propaga-tion,” Physical Review Letters (2010), 10.1103/Phys-RevLett.105.113002.[77] P. Elliott and N.T. Maitra, “Electron correlation via frozengaussian dynamics,” Journal of Chemical Physics (2011),10.1063/1.3630134.[78] K. Chatterjee and K. Pernal, “Excitation energies from ex-tended random phase approximation employed with approxi-mate one- and two-electron reduced density matrices,” Journalof Chemical Physics (2012), 10.1063/1.4766934.[79] D.A. Mazziotti, “Structure of fermionic density matrices:Complete n-representability conditions,” Physical ReviewLetters (2012), 10.1103/PhysRevLett.108.263002.[80] E. A. Burovski, A. S. Mishchenko, N. V. Prokof’ev, and B. V.Svistunov, “Diagrammatic quantum monte carlo for two-bodyproblems: Applied to excitons,” Phys. Rev. Lett. , 186402(2001).[81] Michael Wimmer, S. V. Nair, and J. Shumway, “Biexcitonrecombination rates in self-assembled quantum dots,” Phys.Rev. B , 165305 (2006).[82] U. Woggon, Optical Properties of Semiconductor QuantumDots , Springer Tracts in Modern Physics (Springer Berlin Hei- CONCLUSIONS delberg, 2013).[83] M. Braskan, M. Lindberg, D. Sundholm, and J. Olsen, “Fullconfiguration interaction calculations of electronhole correla-tion effects in strain-induced quantum dots,” physica status so-lidi (b) , 775–779 (2001).[84] S Corni, M Brask´en, M Lindberg, J Olsen, and D Sund-holm, “Size dependence of the electron-hole recombinationrates in semiconductor quantum dots,” Physical Review B -Condensed Matter and Materials Physics , 453131–453139(2003).[85] T. Vanska, M. Lindberg, J. Olsen, and D. Sundholm, “Compu-tational methods for studies of multiexciton complexes,” phys-ica status solidi (b) , 4035–4045 (2006).[86] Tommy V¨ansk¨a and Dage Sundholm, “Interpretation of thephotoluminescence spectrum of double quantum rings,” Phys.Rev. B , 085306 (2010).[87] Dage Sundholm and Tommy Vanska, “Computational meth-ods for studies of semiconductor quantum dots and rings,”Annu. Rep. Prog. Chem., Sect. C: Phys. Chem. , 96–125(2012).[88] T. Juhasz and D.A. Mazziotti, “The cumulant two-particlereduced density matrix as a measure of electron correlationand entanglement,” Journal of Chemical Physics (2006),10.1063/1.2378768.[89] A. J. Coleman, “Density matrices in the quantum theory ofmatter: Energy, intracules and extracules,” International Jour-nal of Quantum Chemistry , 457–464 (1967).[90] J.M. Ugalde, C. Sarasola, L. Domnguez, and R.J. Bovd, “Theevaluation of electronic extracule and intracule densities andrelated probability functions in terms of gaussian basis func-tions,” Journal of Mathematical Chemistry , 51–61 (1991).[91] T. Koga and H. Matsuyama, “Electronic extracule moments ofatoms in position and momentum spaces,” Journal of Chemi-cal Physics , 3424–3430 (1998).[92] P.M.W. Gill, A.M. Lee, N. Nair, and R.D. Adamson, “Insightsfrom coulomb and exchange intracules,” Journal of MolecularStructure: THEOCHEM , 303–312 (2000).[93] P.M.W. Gill, D.P. O’Neill, and N.A. Besley, “Two-electrondistribution functions and intracules,” Theoretical ChemistryAccounts , 241–250 (2003).[94] N.A. Besley, D.P. O’Neill, and P.M.W. Gill, “Computation ofmolecular hartree-fock wigner intracules,” Journal of Chemi-cal Physics , 2033–2038 (2003).[95] P.M.W. Gill, D.L. Crittenden, D.P. O’Neill, and N.A. Besley,“A family of intracules, a conjecture and the electron correla-tion problem,” Physical Chemistry Chemical Physics , 15–25(2006).[96] Michael G. Bayne, John Drogo, and Arindam Chakraborty,“Infinite-order diagrammatic summation approach to the ex-plicitly correlated congruent transformed hamiltonian,” Phys.Rev. A , 032515 (2014).[97] M El-Said, “The ground-state energy of an exciton in aparabolic quantum dot,” Semiconductor Science and Technol-ogy , 272 (1994).[98] W. Que, “Excitons in quantum dots with parabolic confine-ment,” Physical Review B , 11036–11041 (1992).[99] A.V. Nenashev, S.D. Baranovskii, M. Wiemer, F. Jansson,R. Osterbacka, A.V. Dvurechenskii, and F. Gebhard, “The-ory of exciton dissociation at the interface between a con-jugated polymer and an electron acceptor,” Physical ReviewB - Condensed Matter and Materials Physics (2011),10.1103/PhysRevB.84.035210.[100] A. Poszwa, “Relativistic electron confined by isotropicparabolic potential,” Physical Review A - Atomic, Molecular, and Optical Physics (2010), 10.1103/Phys-RevA.82.052110.[101] S. Jaziri, “Effects of electric and magnetic fields on excitonsin quantum dots,” Solid State Communications , 171 – 175(1994).[102] V. Halonen, T. Chakraborty, and P. Pietilinen, “Excitons in aparabolic quantum dot in magnetic fields,” Physical Review B , 5980–5985 (1992).[103] J. Song and S.E. Ulloa, “Magnetic field effects on quantumring excitons,” Physical Review B - Condensed Matter andMaterials Physics , 1253021–1253029 (2001).[104] M. Taut, P. MacHon, and H. Eschrig, “Violation of non-interacting v -representability of the exact solutions of theschrdinger equation for a two-electron quantum dot in a ho-mogeneous magnetic field,” Physical Review A - Atomic,Molecular, and Optical Physics (2009), 10.1103/Phys-RevA.80.022517.[105] A.R. Kolovsky, F. Grusdt, and M. Fleischhauer, “Quantumparticle in a parabolic lattice in the presence of a gauge field,”Physical Review A - Atomic, Molecular, and Optical Physics (2014), 10.1103/PhysRevA.89.033607.[106] A.H. Trojnar, E.S. Kadantsev, M. Korkusiski, and P. Hawry-lak, “Theory of fine structure of correlated exciton states inself-assembled semiconductor quantum dots in a magneticfield,” Physical Review B - Condensed Matter and MaterialsPhysics (2011), 10.1103/PhysRevB.84.245314.[107] Wenfang Xie, “Effect of an electric field and nonlinear opticalrectification of confined excitons in quantum dots,” physicastatus solidi (b) , 2257–2262 (2009).[108] Lili He and Wenfang Xie, “Effects of an electric field onthe confined hydrogen impurity states in a spherical parabolicquantum dot,” Superlattices and Microstructures , 266 – 273(2010).[109] L. Fernandez, Y.ndez-Menchero and H.P. Summers, “Stark ef-fect in neutral hydrogen by direct integration of the hamilto-nian in parabolic coordinates,” Physical Review A - Atomic,Molecular, and Optical Physics (2013), 10.1103/Phys-RevA.88.022509.[110] Y.N. Fernandez, M.I. Vasilevskiy, C. Trallero-Giner, andA. Kavokin, “Condensed exciton polaritons in a two-dimensional trap: Elementary excitations and shaping bya gaussian pump beam,” Physical Review B - CondensedMatter and Materials Physics (2013), 10.1103/Phys-RevB.87.195441.[111] A.M. Rey, G. Pupillo, C.W. Clark, and C.J. Williams, “Ultra-cold atoms confined in an optical lattice plus parabolic poten-tial: A closed-form approach,” Physical Review A - Atomic,Molecular, and Optical Physics (2005), 10.1103/Phys-RevA.72.033616.[112] S.-S. Kim, S.-K. Hong, and K.-H. Yeon, “Linear opticalproperties of the semiconductor quantum shell,” Physical Re-view B - Condensed Matter and Materials Physics (2007),10.1103/PhysRevB.76.115322.[113] G. Rezaei, B. Vaseghi, and M. Sadri, “External electric fieldeffect on the optical rectification coefficient of an exciton in aspherical parabolic quantum dot,” Physica B: Condensed Mat-ter , 4596 – 4599 (2011).[114] W. Liu, D.N. Neshev, A.E. Miroshnichenko, I.V. Shadrivov,and Y.S. Kivshar, “Bouncing plasmonic waves in half-parabolic potentials,” Physical Review A - Atomic,Molecular, and Optical Physics (2011), 10.1103/Phys-RevA.84.063805.[115] S.A. Blundell and K. Joshi, “Precise correlation energies insmall parabolic quantum dots from configuration interaction,” CONCLUSIONS
Physical Review B - Condensed Matter and Materials Physics (2010), 10.1103/PhysRevB.81.115323.[116] Y. Zhao, P.-F. Loos, and P.M.W. Gill, “Correlation energyof anisotropic quantum dots,” Physical Review A - Atomic,Molecular, and Optical Physics (2011), 10.1103/Phys-RevA.84.032513.[117] K. Teichmann, M. Wenderoth, H. Prser, K. Pierz, H.W. Schu-macher, and R.G. Ulbrich, “Harmonic oscillator wave func-tions of a self-assembled inas quantum dot measured byscanning tunneling microscopy,” Nano Letters , 3571–3575(2013).[118] A.L. Morales, N. Raigoza, C.A. Duque, and L.E. Oliveira,“Effects of growth-direction electric and magnetic fields onexcitons in gaas- ga1-x alx as coupled double quantum wells,”Physical Review B - Condensed Matter and Materials Physics (2008), 10.1103/PhysRevB.77.113309.[119] F.S. Nammas, A.S. Sandouqa, H.B. Ghassib, and M.K. Al-Sugheir, “Thermodynamic properties of two-dimensional few-electrons quantum dot using the static fluctuation approxima-tion (sfa),” Physica B: Condensed Matter , 4671 – 4677 (2011).[120] Lin-Wang Wang and Alex Zunger, “Pseudopotential calcula-tions of nanoscale cdse quantum dots,” Phys. Rev. B , 9579–9582 (1996).[121] S. F. Boys, “Electronic wave functions. i. a general method ofcalculation for the stationary states of any molecular system,”Proceedings of the Royal Society of London. Series A, Math-ematical and Physical Sciences , 542–554 (1950).[122] K. Singer, “The use of gaussian (exponential quadratic) wavefunctions in molecular problems. i. general formulae for theevaluation of integrals,” Proceedings of the Royal Society ofLondon. Series A. Mathematical and Physical Sciences ,412–420 (1960).[123] B. Joakim Persson and Peter R. Taylor, “Accurate quantum-chemical calculations: The use of gaussian-type geminal func-tions in the treatment of electron correlation,” The Journal ofChemical Physics , 5915–5926 (1996).[124] B. Joakim Persson and Peter R. Taylor, “Molecular inte-grals over gaussian-type geminal basis functions,” TheoreticalChemistry Accounts , 240–250 (1997)., 240–250 (1997).